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- ItemMathematical modelling of In-vivo HIV optimal therapy and management(Strathmore University, 2018) Ngina, Purity MuthoniHuman Immunodeficiency Virus (HIV) remains the main cause of premature death globally. In 2013, Kenya was the fourth largest endemic HIV country in the world having over 1.6 million people living with the virus. The fact that there is an increase in the rate of new-HIV infections in Africa and especially in Kenya underscores the need for adequate strategies to cope with this deadly disease and to achieve vision 2020. Where the government envision that, by 2020, 90% of all people living with HIV will know their HIV status, 90% of all people with diagnosed HIV infection will receive sustained antiretroviral therapy and 90% of all people receiving antiretroviral therapy will have viral suppression. Currently, there is no known cure for HIV, hence the most optimal way is the management of HIV infected people to prevent virus progression and HIV transmission. Although there has been progress in management of HIV by the use of antiretroviral drugs (ARTs), long-term use of these ARTs leads to overwhelming challenges. These challenges are: toxicity of the medication, nonadherence problems as a result of inaccessibility of comprehensive care centres, drug resistance-mutations and significant financial burdens. This study aimed at formulating and analysing mathematical in-vivo models for the interaction between HIV virions, CD4+ T-cells, CD8+ T-cells and the optimal control for effective therapy, whose numerical simulations would assist in giving more insight about the challenges aforementioned.Various mathematical methods including ordinary differential equations, Runge-Kutta forth order scheme and optimal control theory have been applied in the development and the analysis of the model. Analysis of the formulated model indicates existence of multiple equilibria whose stability and bifurcation analysis have been presented. From the simulated results, we have noted that early initiation of HIV treatment reduce viral replication in HIV infected people. In particular, highly active antiretroviral therapy (HAART) which include the combination therapy of Fusion inhibitor (FI), Reverse Transcriptase inhibitor (RTI) and Protease inhibitor (PI) in different proportions have been found to be more effective in treating HIV than a single drug therapy. The model simulations show how to best choose the proportions of FI, RTI and PI in order to maintain an acceptable level of CD4+ T-cells and, at the same time, reduce the side effects associated with their long term use. In addition, the most optimal way of administering ART drugs that lead to maximum benefit has been predicted from optimal control simulation. The findings give a significant explanation of why late initiation of ARTs might not be helpful to an HIV infected person and suggest that the controls ought to be optimal at the acute phase of infection where the viral replication is extremely high. If the controls are well implemented, many potential infections would be averted by lowering the viral load and increasing the number of the T-helper cells. This, in turn, will also lead to reduction in HIV transmission. Therefore, there is need for increased awareness campaigns to encourage people to know their HIV status and adhere to the prescribed treatment.The research outcomes in this study emphasizes the importance of “Anza Sasa”campaign that was launched on 15th July 2016 by the Government of Kenya through the Ministry of Health in collaboration with the National AIDS and STI Control program.
- ItemConstruction of a zero-coupon yield curve for the Nairobi Securities Exchange and its application in pricing derivatives(Strathmore University, 2017) Muthoni, LucyYield curves are used to forecast interest rates for different products when their risk parameters are known, to calibrate no-arbitrage term structure models, and (mostly by investors) to detect whether there is arbitrage opportunity. By yield curve information, investors have opportunity of immunizing/hedging their investment portfolios against financial risks if they have to make an investment with some determined time of maturity. Private sector firms look at yields of different maturities and then choose their borrowing strategy. The differences in yields for long maturity and short maturities are an important indicator for central bank to use in monetary policy process. These differences may show the tightness of the government monetary policy and can be monitored to predict recession in coming years. A lot of research has been done in yield curve modeling and as we will see later in the thesis, most of the models developed had one major shortcoming: non differentiability at the interpolating knot points. The aim of this thesis is to construct a zero coupon yield curve for Nairobi Securities Exchange, and use the risk- free rates to price derivatives, with particular attention given to pricing coffee futures. This study looks into the three methods of constructing yield curves: by use of spline-based models, by interpolation and by using parametric models. We suggest an improvement in the interpolation methods used in the most celebrated spline-based model, monotonicity-preserving interpolation on r(t). We also use operator form of numerical differentiation to estimate the forward rates at the knot points, at which points the spot curve is non-differential. In derivative pricing, dynamical processes (Ito^ processes) are reviewed; and geometric Brownian motion is included, together with its properties and applications. Conventional techniques used in estimation of the drift and volatility parameters such as historical techniques are reviewed and discussed. We also use the Hough Transform, an artificial intelligence method, to detect market patterns and estimate the drift and volatility parameters simultaneously. We look at different ways of calculating derivative prices. For option pricing, we use different methods but apply Bellalahs models in calculation of the Coffee Futures prices because they incorporate an incomplete information parameter.
- ItemA Copula-based approach to differential gene expression analysis(Strathmore University, 2006) Chaba, Linda AkothMicroarray technology has revolutionized genomic studies by enabling the study of differential expression of thousands of genes simultaneously. The main objective in microarray experiments is to identify a panel of genes that are associated with a disease outcome or trait. In this thesis, we develop and evaluate a semi-parametric copula-based algorithm for gene selection that does not depend on the distributions of the covariates, except that their marginal distributions are continuous. A comparison of the developed method with the existing methods is done based on power to identify differentially expressed genes (DEGs) and control of Type I error rate via a simulation study. Simulations indicate that the copula-based model has a reasonable power in selecting differentially expressed gene and has a good control of Type I error rate. These results are validated in a publicly available melanoma dataset. The copula-based approach turns out to be useful in finding genes that are clinically important. Relaxing parametric assumptions on microarray data may yield procedures that have good power for differential gene expression analysis.
- ItemSmooth tests of goodness of FIT for hazard Functions: an application to HIV retention data(Strathmore University, 2017) Odhiambo, Collins Ojwang’In this study, we apply the methodology of smooth tests of goodness-of-fit to hazard functions. The smooth test formulation applied here is an extension of Neyman’s smooth test and is obtained by nesting the null hypothesis in a larger class of probability and hazard rate functions. The study revisits Neyman’s smooth tests and its data-driven versions in the context of classical probability and survival analysis. Though several authors have theoretically looked at the development of Neyman’s smooth tests, the main contribution of this study is modelling loss to follow-up in HIV retention. To the best of our knowledge, this issue has not been given its due share of coverage in the literature. We extend methods proposed by Rayner et al. (2009); Pena (1998a,b) and Kraus (2007a), to an HIV retention setting. The applications dealt with in this thesis also covers performance of other goodness-of-fit (GOF) approaches and compares them with that of smooth tests. Three main methodological approaches are covered under the research methodology. Part I revisits smooth tests for various probability distributions and applies the test when assessing the fit for the two-parameter Weibull distribution to an HIV retention data under the complete and uncensored data scenario. Part II looks at the application of smooth test to Cox proportional hazards models. We assess the proportionality assumption in the two sample problem in cancer survival studies. Part III covers recurrent event situation. We fit Block, Borges and Savits (BBS) minimal repair model to loss to follow-up (LFTU) data and assesses the performance of the smooth test in terms of power.More specifically, Chapter 1 deals with background of GOF in classical probability and survival distributions. The motivation for the study, overview of the smooth test of GOF and comparison with other GOF tests is also covered in this chapter. In Chapter 2, we provide a review of the literature. Chapter 3 details research methodology. We present analysis and results in Chapter 4. Chapter 5 discusses important findings using simulated and real data in the context of HIV retention and overall survival in cancer studies. Chapter 6 covers summary of the thesis, the limitations of the study and possible extensions of the smooth GOF to discrete probability cases. All computations have been implemented in R and the scripts are briefly described in Appendix A. The chapters are self-contained in order to achieve our objective of covering the applications smooth tests of goodness-of-fit approach from distributions with noncensored data to extensions in recurrent events.A major limitation of this study, is that, in clinical studies, particularly involving LTFU data, incomplete data is frequently encountered. Analysis of severity of data incompleteness is a subject of future research.